page_title: Integral
section_title: Multiple integration
hierarchical_section_title: Integral / Extensions / Multiple integration
caption_attribution_description: Illustration of volume under a surface (double integral)
context_page_description: In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus; its inverse operation, differentiation, is the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral
  can be interpreted informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.
  The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
  The integrals discussed in this article are those termed definite integrals.
context_section_description: Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane that contains its domain. For example, a function in two dimensions depends on two real variables, x and y, and the integral of a function f over the rectangle R given as the Cartesian product of two intervals  can be written
  where the differential dA indicates that integration is taken with respect to area. This double integral can be defined using Riemann sums, and represents the (signed) volume under the graph of z = f(x,y) over the domain R. Under suitable conditions (e.g., if f is continuous), Fubini's theorem states that this integral can be expressed as an equivalent iterated integral
  This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over R uses a double integral sign:
  Integration over more general domains is possible. The integral of a function f, with respect to volume, over an n-dimensional region D of  is denoted by symbols such as:
caption: Double integral computes volume under a surface 
    
      
        
          z
          =
          f
          (
          x
          ,
          y
          )
        
      
      {\displaystyle z=f(x,y)}
exif: {}